In What’s the Matter with Kansas, Thomas Frank argues that the Republican Party has
redrawn the landscape of contemporary American politics by displacing the Democratic
Party as the party of the working class. Although others have tested the electoral
implications of Frank’s arguments, no one has examined the phenomenon itself—the
belief that Republicans are the party of the working class. To test this claim, we created a
survey expressly for the purpose of examining the class-based images of the parties.
Following Frank’s argument, we examine whether the public believes Republicans are
the party of the working class and specifically whether the working class and Evangelical
Christians are most likely to hold this belief. Contrary to Frank’s argument, the
Democrats are the party most Americans associate with the working class. Furthermore,
we find that the working class and Evangelical Christians are no more likely to hold this
belief than respondents in other social classes or faiths.

Unfortunately, these critiques will reach .1% of Frank's audience, which will never know how little empirical support his claims actually have.

I'm not sure this gets to the heart of Frank's argument. As I understand it, he is not arguing that voters view the Republican party as a champion of the working class, but rather that working class voters are voting for Republicans even though the Democratic party better represents their interests. Instead they are voting on the basis of moral convictions, shared values, etc.